Cohomology of arbitrary spin currents in AdS3

We study conserved currents of any integer or half-integer spin built from massless scalar and spinor fields inAdS ₃. We show that 2-forms dual to the conserved currents inAdS ₃ are exact in the class of infinite expansions in higher derivatives of the matter fields with the coefficients containing inverse powers of the cosmological constant. This property has no analogue in the flat space and may be related to the holography of the AdS spaces. “Improvements” to the physical currents are described as the trivial local current cohomology class. A complex (T^S, D) of spin-s currents is defined, and the cohomology group H¹(T^S, D)= ℂ^2s+1is found. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 3–25, April, 2000.     

Functional representations for correlators of spin chains

For a finite spin chain, a functional representation of the generating function of the XY-model at nonzero temperature is obtained. Bibliography: 16 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 192–207. Translated by V. S. Kapitonov.     

Quantum solitons and the Haldane phase in antiferromagnetic spin chains

A survey is given of the relevance of solitons for the present understanding of antiferromagnetic quantum spin chains. For the -chain the basic elementary excitations are solitons. Two solitons combine to form a magnon and its internal degree of freedom is observed as an excitation continuum. For S = 1 chains the existence of the massive Haldane phase is related to the dissociation of soliton pairs (pairs of domain walls with respect to antiferromagnetic order). Based on the hidden (or string) order, the appropriate mean field theory for antiferromagnetic S = 1 chains is introduced as based on uncorrelated solitons and applied to calculate correlation functions. It is described how to include correlations between solitons using an approximate mapping to an effective chain. The basic elementary excitations in the Haldane phase are identified as solitons with respect to the hidden order.     

Darboux-Nijenhuis variables for open generalized Toda chains

We consider the possibility of using the Sklyanin method to construct Darboux-Nijenhuis variables of special form in the example of generalized open Toda chains associated with classical root systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 440–456, September, 2007.     

Stability of the antiferromagnetic state of finite spin chains

The magnetic field strength at which the antiferromagnetic ordering becomes unstable is found analytically for a model of a one-dimensional ferromagnet with classical spin and anisotropic interaction. Such instabilities are inherent for the surface spin-flop transition, which is actively studied in hyperlattices. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115, No. 2, pp 275–279, May, 1998     

Observer design for open and closed trophic chains

Monitoring of ecological systems is one of the major issues in ecosystem research. The concepts and methodology of mathematical systems theory provide useful tools to face this problem. In many cases, state monitoring of a complex ecological system consists in observation (measurement) of certain state variables, and the whole state process has to be determined from the observed data. The solution proposed in the paper is the design of an observer system, which makes it possible to approximately recover the state process from its partial observation. Such systems-theoretical approach has been applied before by the authors to Lotka–Volterra type population systems. In the present paper this methodology is extended to a non-Lotka–Volterra type trophic chain of resource–producer–primary consumer type and numerical examples for different observation situations are also presented.     

Jordanian deformation of the open XXX spin chain

We find the general solution of the reflection equation associated with the Jordanian deformation of the SL(2)-invariant Yang R-matrix. A special scaling limit of the XXZ model with general boundary conditions leads to the same K-matrix. Following the Sklyanin formalism, we derive the Hamiltonian with the boundary terms in explicit form. We also discuss the structure of the spectrum of the deformed XXX model and its dependence on the boundary conditions.     

The number of open chains of length three and the parity of the number of open chains of length k in self-complementary graphs

In this paper it is shown, using the Sieve formula, that the number of open chains of length k, k ? 5, in a self-complementary (s.c.) graph is always even. As a corollary, it follows that the number of hamiltonian chains in a s.c. graph of order p > 5 is even, a result proved earlier by Camion. Further, the minimum number and the maximum number of open chains of length 3 in s.c. graphs of order p are determined, and the s.c. graphs of order p which attain these bounds are characterized.     

The commutativity of integrals of motion for quantum spin chains and elliptic functions identities

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.      

Matchings in pentagonal chains

An explicit recurrence is derived for the matching polynomial of the pentagonal chain. From this, explicit formulae are obtained for the number of defect-d matchings in the pentagonal chain, for various values of d.     

Persistent equilibria in strategic games

A perfect equilibrium [Selten] can be viewed as a Nash equilibrium with certain properties of local stability. Simple examples show that a stronger notion of local stability is needed to eliminate unreasonable Nash equilibria. The persistent equilibrium is such a notion. Properties of this solution are studied. In particular, it is shown that in each strategic game there exists a pesistent equilibrium which is perfect and proper.     

Hamiltonian chains in hypergraphs

A cyclic ordering of the vertices of a k‐uniform hypergraph is called a hamiltonian chain if any k consecutive vertices in the ordering form an edge. For k = 2 this is the same as a hamiltonian cycle. We consider several natural questions about the new notion. The main result is a Dirac‐type theorem that provides a sufficient condition for finding hamiltonian chains in k‐uniform hypergraphs with large (k − 1)‐minimal degree. If it is more than than the hypergraph contains a hamiltonian chain. © 1999 Wiley & Sons, Inc. J Graph Theory 30: 205–212, 1999     

Matchings in benzene chains

An explicit recurrence relation is derived for the matching polynomial of the general benzene chain B_n. A table is given for chains of length up to 6. Explicit formulae are then obtained for the first five coefficients of the matching polynomial of B_n. Finally, results are deduced for the number of perfect matchings and the number of matchings with two nodes.     

Principal currents

An operator theoretic analog of a local Chern class is discussed for certain finitely generated operator algebras. This gives rise to an index theory for the maximal ideals of the algebra. The algebraic receiver for this index is the Grothendieck group of the algebra completed at a maximal ideal.This research was partially supported by grants from the National Science Foundation.